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Let $B$ be a block algebra of a group algebra $FG$ of a finite group $G$ over a field $F$ of characteristic $p>0$. This paper studies ring theoretic properties of the representation ring $T^\Delta(B,B)$ of perfect $p$-permutation $(B,B)$-bimodules and properties of the $k$-algebra $k\otimes_\mathbb{Z} T^\Delta(B,B)$, for a field $k$. We show that if the Cartan matrix of $B$ has $1$ as an elementary divisor then $[B]$ is not primitive in $T^\Delta(B,B)$. If $B$ has cyclic defect groups we determine a primitive decomposition of $[B]$ in $T^\Delta(B,B)$. Moreover, if $k$ is a field of characteristic different from $p$ and $B$ has cyclic defect groups of order $p^n$ we describe $k\otimes_\mathbb{Z} T^\Delta(B,B)$ explicitly as a direct product of a matrix algebra and $n$ group algebras., Comment: 16 pages

The goal of a series of papers is to define $G$-actions on various $A$-fibered structures, where $G$ is a finite group and $A$ is an abelian group. One prominent such example is the $A$-fibered Burnside ring. If $A=\mathbb{C}^\times$, it is also called the ring of monomial representations (introduced by Dress in \cite{Dress1971}) and is the natural home for the canonical induction formula (see \cite{Boltje1990}). In this first part of the series, motivated by constructions in \cite{BoucMutlu}, we introduce $A$-fibered structures on posets, on abstract simplicial complexes, and on $A$-bundles over topological spaces, together with natural notions of homotopy, and functors between these structures respecting homotopy. In a sequel we will continue with $G$-representations in these $A$-fibered structures and associate to them elements in the $A$-fibered Burnside ring.

Let $G$ and $H$ be finite groups. We give a condition on $G$ and $H$ that implies that the $A$-fibered Burnside rings $B^A(G)$ and $B^A(H)$ are isomorphic. As a consequence, we show the existence of non-isomorphic groups $G$ and $H$ such that $B^A(G)$ and $B^A(H)$ are isomorphic rings. Here, the abelian fiber group $A$ can be chosen in a non-trivial way, that is, such that $B^A(G)$ and $B^A(H)$ are strictly bigger than the Burnside rings of $G$ and $H$, for which such counterexamples are already known.

In this article, we propose a concept of anti-involution for Green biset functors and we provide equivalent definitions. We present $\star$-Green biset functors and we study their orthogonal units and the orthogonal automorphisms in their associated categories.

Let $G$ be a finite group, $p$ a prime, and $(K,\mathcal{O},F)$ a $p$-modular system. We prove that the trivial source ring of $\mathcal{O} G$ is isomorphic to the ring of {\em coherent} $G$-stable tuples $(\chi_P)$, where $\chi_P$ is a virtual character of $K[N_G(P)/P]$, $P$ runs through all $p$-subgroups of $G$, and the coherence condition is the equality of certain character values. We use this result to describe the group of orthogonal units of the trivial source ring as the product of the unit group of the Burnside ring of the fusion system of $G$ with the group of coherent $G$-stable tuples $(\varphi_P)$ of homomorphisms $N_G(P)/P\to F^\times$. The orthogonal unit group of the trivial source ring of $\mathcal{O} G$ is of interest, since it embeds into the group of $p$-permutation autoequivalences of $\mathcal{O} G$., Comment: 18 pages, corrected some typos, to appear in Journal of Algebra

We give sufficient conditions on $p$-blocks of $p$-nilpotent groups over $\mathbb{F}_p$ to be splendidly Rickard equivalent and $p$-permutation equivalent to their Brauer correspondents. The paper also contains Galois descent results on $p$-permutation modules and $p$-permutation equivalences that hold for arbitrary groups., Comment: 15 pages, some rearrangement of content of Section 5. Clarification of some arguments

A perfect isometry $I$ (introduced by Brou\'e) between two blocks $B$ and $C$ is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character $\psi$ of $C$ to $\pm$ an irreducible character of $B$. Brou\'e proved that the ratio of the codegrees of $\psi$ and $I(\psi)$ is a rational number with $p$-value zero and that its class in $\mathbb{F}_p$ is independent of $\psi$. We call this element the Brou\'e invariant of $I$. The goal of this paper is to show that if $I$ comes from a $p$-permutation equivalence or a splendid Rickard equivalence between $B$ and $C$ then, up to a sign, the Brou\'e invariant of $I$ is determined by local data of $B$ and $C$ and therefore, up to a sign, is independent of the $p$-permutation equivalence or splendid Rickard equivalence. Apart from results on $p$-permutation equivalences, our proof requires new results on extended tensor products and bisets that are also proved in this paper. As application of the theorem on the Brou\'e invariant we show that various refinements of the Alperin-McKay Conjecture, introduced by Isaacs-Navarro, Navarro, and Turull are consequences of $p$-permutation equivalences or Rickard equivalences over a sufficiently large complete discrete valuation ring or over $\mathbb{Z}_p$, depending on the refinement., Comment: 24 pages, some typos corrected from version 1. Most importantly, Section 5 added, containing an application to various refinements of the Alperin-McKay conjecture due to Isaacs-Navarro, Navarro, and Turull

We extend the notion of a {$p$-permutation equivalence} between two $p$-blocks $A$ and $B$ of finite groups $G$ and $H$, from the definition in [Boltje-Xu 2008] to a virtual $p$-permutation bimodule whose components have twisted diagonal vertices. It is shown that various invariants of $A$ and $B$ are preserved, including defect groups, fusion systems, and K\"ulshammer-Puig classes. Moreover it is shown that $p$-permutation equivalences have additional surprising properties. They have only one constituent with maximal vertex and the set of $p$-permutation equivalences between $A$ and $B$ is finite (possibly empty). The paper uses new methods: a consequent use of module structures on subgroups of $G\times H$ arising from Brauer constructions which in general are not direct product subgroups, the necessary adaptation of the notion of tensor products between bimodules, and a general formula (stated in these new terms) for the Brauer construction of a tensor product of $p$-permutation bimodules., Comment: 48 pages

Let $A$ be an abelian group such that $\mathrm{Hom}(G,A)$ is finite for all finite groups $G$, and let $\mathbb{K}$ be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in $A$. In this paper we prove foundational properties of the $A$-fibered Burnside ring functor $B_{\mathbb{K}}^A$ as an $A$-fibered biset functor over $\mathbb{K}$. This includes the determination of the lattice of subfunctors of $B_{\mathbb{K}}^A$ and the determination of the composition factors of $B_{\mathbb{K}}^A$. The results of the paper extend results of Co\c{s}kun and Y\i lmaz for the $A$-fibered Burnside ring functor restricted to $p$-groups and results of Bouc in the case that $A$ is trivial, i.e., the case of the Burnside ring functor over fields of characteristic zero., Comment: 26 pages, referee comments incorporated, consolidated prerequisites from previous Sections 2--4 to Section 2, changed abstract slightly, accepted by Algebras and Representation Theory

To any block idempotent $b$ of a group algebra $kG$ of a finite group $G$ over a field $k$ of characteristic $p>0$, Puig associated a fusion system and proved that it is saturated if the $k$-algebra $kC_G(P)e$ is split, where $(P,e)$ is a maximal $kGb$-Brauer pair. We investigate in the non-split case how far the fusion system is from being saturated by describing it in an explicit way as being generated by the fusion system of a related block idempotent over a larger field together with a single automorphism of the defect group., Comment: 11 pages

In this article we define the $-_+$-construction and the $-^+$-construction, that was crucial in the theory of canonical induction formulas (see \cite{Boltje1998b}), in the setting of biset functors, thus providing the necessary framework to define and construct canonical induction formulas for representation rings that are most naturally viewed as biset functors. Additionally, this provides a unified approach to the study of a class of functors including the Burnside ring, the monomial Burnside ring and global representation ring., Comment: 21 pages

We show that the subgroup of the Picard group of a $p$-block of a finite group given by bimodules with endopermutation sources modulo the automorphism group of a source algebra is determined locally in terms of the fusion system on a defect group. We show that the Picard group of a block over the a complete discrete valuation ring ${\mathcal O}$ of characteristic zero with an algebraic closure $k$ of ${\mathbb F}_p$ as residue field is a colimit of finite Picard groups of blocks over $p$-adic subrings of ${\mathcal O}$. We apply the results to blocks with an abelian defect group and Frobenius inertial quotient, and specialise this further to blocks with cyclic or Klein four defect groups.

The theory of biset functors, introduced by Serge Bouc, gives a unified treatment of operations in representation theory that are induced by permutation bimodules. In this paper, by considering fibered bisets, we introduce and describe the basic theory of fibered biset functors which is a natural framework for operations induced by monomial bimodules. The main result of this paper is the classification of simple fibered biset functors., Comment: 56 pages

Let $k$ be a field of characteristic $0$, let $\mathsf{C}$ be a finite split category, let $\alpha$ be a 2-cocycle of $\mathsf{C}$ with values in the multiplicative group of $k$, and consider the resulting twisted category algebra $A:=k_\alpha\mathsf{C}$. Several interesting algebras arise that way, for instance, the Brauer algebra. Moreover, the category of biset functors over $k$ is equivalent to a module category over a condensed algebra $\varepsilon A\varepsilon$, for an idempotent $\varepsilon$ of $A$. In [2] the authors proved that $A$ is quasi-hereditary (with respect to an explicit partial order $\le$ on the set of irreducible modules), and standard modules were given explicitly. Here, we improve the partial order $\le$ by introducing a coarser order $\unlhd$ leading to the same results on $A$, but which allows to pass the quasi-heredity result to the condensed algebra $\varepsilon A\varepsilon$ describing biset functors, thereby giving a different proof of a quasi-heredity result of Webb, see [26]. The new partial order $\unlhd$ has not been considered before, even in the special cases, and we evaluate it explicitly for the case of biset functors and the Brauer algebra. It also puts further restrictions on the possible composition factors of standard modules., Comment: 39 pages

The bifree double Burnside ring $B^\Delta(G,G)$ of a finite group $G$ has a natural anti-involution. We study the group $B^\Delta_\circ(G,G)$ of orthogonal units in $B^\Delta(G,G)$. It is shown that this group is always finite and contains a subgroup isomorphic to $B(G)^\times\rtimes \Out(G)$, where $B(G)^\times$ denotes the unit group of the Burnside ring of $G$ and $\Out(G)$ denotes the outer automorphism group of $G$. Moreover it is shown that if $G$ is nilpotent then $B^\Delta_\circ(G,G)\cong B(G)^\times\rtimes \Out(G)$. The results can be interpreted as positive answers to questions on equivalences of $p$-blocks of group algebras in the case that the block is the group algebra of a $p$-group., Comment: 15 pages, minor changes, final version, to appear in Journal of Pure and Applied Algebra

We determine the blocks, i.e., the primitive central idempotents, of the bifree double Burnside ring $B^\Delta(G,G)$ and the left-free double Burnside ring $B^{\trl}(G,G)$, as well as the primitive central idempotents of the algebras arising from scalar extension to $\QQ$., Comment: 24 pages; minor changes; final version; to appear in Israel Journal of Mathematics

A category is called {\em split} if for every morphism $s\colon X\to Y$ there exists a morphism $t\colon Y\to X$ such that $s\circ t\circ s = s$. Let $C$ be a finite split category, let $k$ be a field of characteristic 0 and let $\alpha$ be a 2-cocycle of $C$ with values in the unit group of $k$. Then the twisted category algebra $k_\alpha C$ is a quasi-hereditary algebra., Comment: 11 pages. Added Section 4 on construction of standard modules. Final version

In [Boltje,Hartmann: Permutation resolutions for Specht modules, J. Algebraic Combin. 34 (2011), 141-162], a chain complex was constructed in a combinatorial way which conjecturally is a resolution of the (dual of the) integral Specht module for the symmetric group in terms of permutation modules. In this paper we extend the definition of the chain complex to the integral Iwahori Hecke algebra and prove the same partial exactness results that were proved in the symmetric group case. A complete proof of the exactness conjecture in the symmetric group case was recently given by Santana and Yudin, Adv. in Math. 229 (2012), 2578-2601., Comment: 23 pages, fixed mistake in proof of Prop. 2.2, other minor changes, to appear in Journal of Algebra

For a finite group $G$, we introduce a multiplication on the $\QQ$-vector space with basis $\scrS_{G\times G}$, the set of subgroups of $G\times G$. The resulting $\QQ$-algebra $\Atilde$ can be considered as a ghost algebra for the double Burnside ring $B(G,G)$ in the sense that the mark homomorphism from $B(G,G)$ to $\Atilde$ is a ring homomorphism. Our approach interprets $\QQ B(G,G)$ as an algebra $eAe$, where $A$ is a twisted monoid algebra and $e$ is an idempotent in $A$. The monoid underlying the algebra $A$ is again equal to $\scrS_{G\times G}$ with multiplication given by composition of relations (when a subgroup of $G\times G$ is interpreted as a relation between $G$ and $G$). The algebras $A$ and $\Atilde$ are isomorphic via M\"obius inversion in the poset $\scrS_{G\times G}$. As an application we improve results by Bouc on the parametrization of simple modules of $\QQ B(G,G)$ and also of simple biset functors, by using results by Linckelmann and Stolorz on the parametrization of simple modules of finite category algebras. Finally, in the case where $G$ is a cyclic group of order $n$, we give an explicit isomorphism between $\QQ B(G,G)$ and a direct product of matrix rings over group algebras of the automorphism groups of cyclic groups of order $k$, where $k$ divides $n$., Comment: 41 pages. Changed title from "Ghost algebras of double Burnside algebras via Schur functors" and other minor changes. Final version

For a finite group $G$, we define a ghost ring and a mark homomorphism for the double Burnside ring of left-free $(G,G)$-bisets. In analogy to the case of the Burnside ring $B(G)$, the ghost ring has a much simpler ring structure, and after tensoring with $\QQ$ one obtains an isomorphism of $\QQ$-algebras. As an application of a key lemma, we obtain a very general formula for the Brauer construction applied to a tensor product of two $p$-permutation bimodules $M$ and $N$ in terms of Brauer constructions of the bimodules $M$ and $N$. Over a field of characteristic 0 we determine the simple modules of the left-free double Burnside algebra and prove semisimplicity results for the bifree double Burnside algebra. These results carry over to results about biset-functor categories. Finally, we apply the ghost ring and mark homomorphism to fusion systems on a finite $p$-group. We extend a remarkable bijection, due to Ragnarsson and Stancu, between saturated fusion systems and certain idempotents of the bifree double Burnside algebra over $\ZZ_{(p)}$, to a bijection between all fusion systems and a larger set of idempotents in the bifree double Burnside algebra over $\QQ.$, Comment: 43 pages, minor changes of previous version, to appear in "Advances in Mathematics"

In this note we present an algorithm for the construction of the unit group of the Burnside ring $\Omega(G)$ of a finite group $G$ from a list of representatives of the conjugacy classes of subgroups of G., Comment: corrected and slightly extended version of the article published in: Groups St. Andrews 2005. Vol. 1, 230--236, London Math. Soc. Lecture Note Ser., 339, Cambridge Univ. Press, Cambridge, 2007